Download Solving Trig Equations

Math 1060
Solving Trigonometric Equations
Solving Trigonometric Equations
The easiest trig equations just involve a good knowledge of the unit circle.
√
1. Find a value for x such that sin(x) = −
2
2 .
2. Find a value for θ such that cos(θ) = 12 .
√
3. Find a value for t such that tan(t) = − 3.
In the above, you found a solution to those equations. When dealing with trig functions, however,
there may be more than one solution. In fact, there’s usually an infinite number of solutions. Given
an angle θ, we can write all angles that are coterminal with θ as “θ + 2πk, for any integer k.” For
example, if we want to represent the set of angles {0, 2π, 4π, 6π, −2π, −4π, . . .}, we could just write
“0 + 2πk, k ∈ Z” (that “k ∈ Z” stuff is mathematician shorthand for “k is any integer.”).
√
4. Find all values of x such that sin(x) = −
2
2 .
5. Find all values of t such that tan(t) = 1.
6. Find all values of θ such that csc(θ) = 1.
If you have a more complicated trig equation, your main goal is to use algebraic techniques to
transform it into something simple, like one of those above.
√
7. Solve for t: 2 cos t = −1.
Steven Pon
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rev. September 30, 2014
Math 1060
8. Solve for t:
Solving Trigonometric Equations
3 + 2 sin t
= sin t.
5
Sometimes we get tired of writing +2πk all the time. A common thing to do is to restrict our
attention to solutions that lie in the interval [0, 2π).
9. Find all solutions in the interval [0, 2π): 1 =
10. Find all solutions in the interval [0, 2π):
1 + 3 cos θ
.
5 cos θ − 2
6 sec t + 2
= 2.
2 sec t − 1
Sometimes, some more complicated algebraic techniques might be required. Things like factoring,
and then using the fact that AB = 0 =⇒ A = 0 or B = 0. Things like using the fact that
sin(x)
1
sec(x) = cos(x)
, or tan(x) = cos(x)
. Things like treating sin(x) as a single “thing” (which it is), and
2
factoring sin (x) − 2 sin(x) − 3 exactly the same way you would factor u2 − 2u − 3.
√
11. Find all solutions in [0, 2π): 2 sin2 t + 3 sin t = 0 (Try factoring the left hand side.)
Steven Pon
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rev. September 30, 2014
Math 1060
Solving Trigonometric Equations
12. Find all solutions: 2 sin t cos t = sin t (Try moving all terms to one side and then factoring.)
13. Find all solutions in [0, 2π): 2 cos2 t + cos t − 1 = 0. (Try factoring it like a quadratic.)
14. Find all solutions in [0, 2π): sin t + tan t = 0. (Try rewriting tan(x), then factoring.)
15. Solve for θ: 2 sin2 θ − 3 sin θ + 1 = 0
16. Solve for x: tan x sec x +
Steven Pon
√
2 tan x = 0
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rev. September 30, 2014
Math 1060
Solving Trigonometric Equations
Sometimes your answers have to be expressed using inverse trig functions, since they won’t always
work out nicely.
17. Find two solutions for x: 3 cos2 (x) + cos(x) − 2 = 0.
What if you had a more complicated expression inside a trig function? Something like tan(2x)?
Hint: Let u = 2x, solve for u, and then substitute back to solve for x.
18. Find all solutions in [0, 2π): tan( x2 ) = 1.
√
19. Find all solutions: cos(2x) = −
Steven Pon
2
2 .
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Math 1060
Solving Trigonometric Equations
You can also use trig identities to help out with simplifying equations.
20. Find all solutions to sin(2x) = cos x.
21. Solve sec2 x − 2 tan x = 4.
22. Find all solutions in [0, 2π) of 2 cot2 (x) + csc2 (x) − 2 = 0.
Steven Pon
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rev. September 30, 2014